Two sound sources

Where is each one heard louder?

Posted by Łukasz Bolikowski on October 13, 2014

I was sitting in a barber shop for kids the other day.
Two children were being cut, and they were watching different cartoons on TVs in front of them.
In the figure below, children A and B are at (0, 0.5) and (1, 0.5)
and their TVs are at (0, 0) and (1, 0) respectively. I’m sitting at (2, 1).

The two sources of sound and the three listeners.

Now, I heard TV A considerably louder,
even though it was farther away from me than TV B.
However, Kid B seemed to be content with the volume,
i.e., apparently he heard TV B louder than TV A.
It didn’t seem intuitive to me, so I started wondering
(as I didn’t have much else to do)
what is the shape of the area in which TV B is heard louder.

In more abstract terms:
let’s have two sound sources A and B,
which are d units apart.
Let A be k times stronger.
What is the shape of the area where B is heard louder?

The result is quite easy to derive
(I was almost able to get it right in my head without pencil and paper)
and — at least for me — a bit counter-intuitive,
because V jnfa’g rkcrpgvat gur nern gb or svavgr.